Measuring Large Optical Transmission Matrices of
Disordered Media
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Citation
Yu, Hyeonseung, Timothy R. Hillman, Wonshik Choi, Ji Oon Lee,
Michael S. Feld, Ramachandra R. Dasari, and YongKeun Park.
“Measuring Large Optical Transmission Matrices of Disordered
Media.” Physical Review Letters 111, no. 15 (October 2013). ©
2013 American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevLett.111.153902
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PHYSICAL REVIEW LETTERS
PRL 111, 153902 (2013)
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11 OCTOBER 2013
Measuring Large Optical Transmission Matrices of Disordered Media
Hyeonseung Yu,1 Timothy R. Hillman,2 Wonshik Choi,3 Ji Oon Lee,4 Michael S. Feld,2,*
Ramachandra R. Dasari,2 and YongKeun Park1,†
1
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea
G. R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
3
Department of Physics, Korea University, Seoul 136-701, South Korea
4
Department of Mathematical Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea
(Received 4 June 2013; published 10 October 2013)
2
We report a measurement of the large optical transmission matrix (TM) of a complex turbid medium.
The TM is acquired using polarization-sensitive, full-field interferometric microscopy equipped with a
rotating galvanometer mirror. It is represented with respect to input and output bases of optical modes,
which correspond to plane wave components of the respective illumination and transmitted waves. The
modes are sampled so finely in angular spectrum space that their number exceeds the total number of
resolvable modes for the illuminated area of the sample. As such, we investigate the singular value
spectrum of the TM in order to detect evidence of open transmission channels, predicted by randommatrix theory. Our results comport with theoretical expectations, given the experimental limitations of the
system. We consider the impact of these limitations on the usefulness of transmission matrices in optical
measurements.
DOI: 10.1103/PhysRevLett.111.153902
PACS numbers: 42.25.Dd, 05.60.Cd, 73.23.b
Optical wave propagation through highly scattering
media is a fundamental physical phenomenon relevant to
numerous fields including imaging through turbid media
[1,2] and quantum information processing [3]. The scattering matrix is the linear transformation relating two
monochromatic, propagating complex wave fields: that
incident upon the sample; and that elastically scattered
from it. The samples we consider are broad in lateral extent
and are bounded (in the axial direction) by parallel-plane
interfaces. Thus, all of the light may be regarded as entering and exiting the sample through the interfaces. The
transmission matrix (TM) is a submatrix of scattering
matrix, which describes propagation through the sample
in a particular direction—light enters the sample at the
‘‘front’’ interface and exits (is scattered from) the ‘‘back’’
interface. The TM is generally expressed with respect to
bases of the normal modes of the respective incident and
scattered wave fields [4]. Each mode Ei is the complexamplitude representation of the optical field corresponding
to one particular polarization and propagation direction.
The TM is therefore defined as the matrix t, with complexvalued entries tij connecting the incident optical field at the
ith output mode to the jth input mode,
Eout
i ¼
N
X
tij Ein
j :
(1)
j¼1
Common optical elements such as lenses or polarizers
have simple TMs with low information content, as suggested by the fact that their effects on an incident wave
field can be described using a simple equation. By contrast,
disordered complex media such as paint layers or
0031-9007=13=111(15)=153902(5)
biological tissues have TMs with high information content;
the enormous number of elements are apparently independent [5]. However, subtle correlations between the TM
elements exist that can be discerned using statistical analysis. These correlations can give important insights into the
mesoscopic properties of a complex medium. Additionally,
knowledge of the TM can be used for imaging and
focusing applications in environments with multiple light
scattering [2,6–9].
The TM is a key concept in mesoscopic transport
theory [4], and it has been directly measured in the microwave regime [10]. Previous, intensity-based experimental
methods for considering light transport through complex
media include: speckle pattern correlation [11], coherent
backscattering [12,13], and vector transmission matrix
measurement [14]. A significant breakthrough in opticalregime TM measurement has been achieved by using a
phase-shifting interferometric system [15,16]. Although
the number of measured TM matrix elements measured
in these experiments was substantial (65 536 elements in
the TM; 256 illumination and 256 detection modes in
Ref. [15], for example), the full TM corresponding to
the specified illuminated area of the sample is many
times larger. Consequently, measurements of optical TMs
reported so far do not seem to exhibit intrinsic correlations
between the modes. This inference can be drawn from the
fact that the distribution of singular values of the measured
matrices obeys a quarter-circle law [15]. In fact, the number of resolvable modes, N, for either the incident or
transmitted far-field illumination depends on the illuminated area of the sample, A, according to the equation
N ¼ 2A=2 , where is the wavelength of the incident
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Ó 2013 American Physical Society
light in the medium surrounding the sample. This equation
accounts for the two orthogonal polarization states of the
propagating wave field [17]. The modes can correspond to
different plane wave angular (k vector) components of
the respective beams (our choice, hereafter), or alternatively, different diffraction-limited spots at the respective
interfaces.
In this Letter, we report a measurement of the large TM
for a highly scattering medium. It was enabled by using a
polarization-sensitive, full-field Mach-Zehnder interferometric microscope equipped with a fast two-axis rotating
galvanometer mirror system. We measured the TM of an
18 18 m2 area of the medium, acquiring 21 078 21 078 entries, that is, a 420 million-element array. We
investigate the singular-value and eigenvalue distribution
of the TM in the context of random-matrix theory (RMT)
developed by Dorokhov, Mello, Perreira, and Kumar
(DMPK) [18,19]. Additionally, we consider the effect of
the limited system numerical aperture (NA) in excluding
the number of modes accessible to the system.
The experimental setup is shown in Fig. 1. A detailed
description of the setup is discussed in Supplemental
Material [20]. A plane wave beam from a linearly polarized
He-Ne laser is first split into a sample and a reference arm.
The two-axis rotating galvonometer mirror system controls
the illumination angle of the beam impinging upon the
scattering sample. The transmitted scattered beam is projected onto a camera, where the scattered beam interferes
with a slightly tilted reference beam to generate off axis
holograms for each illumination angle. Half-wave plates
are located in both the sample and reference arms in order
to control the axes of polarization of the beams.
The sample under study consists of a disordered layer
of spray-painted ZnO nanoparticles with a thickness of
L ¼ 10:5 m, sandwiched between two standard microscope cover slides. The layers were characterized through
measurements of both their total and their angularly
resolved transmission. From these measurements, we
Lens
Beam
splitter
2D rotating mirror
FIG. 1 (color online). Experimental setup. (Inset) scanning
electron micrograph of a ZnO sample (cross section).
(b)
1
1
λ/2 waveplate
Objective lens
100x NA=1.4
21078
output mode
10539
He-Ne
laser
input mode
10539
21078
5 µm
1
1
Turbid
Medium
(a)
output mode
10539
Beam
splitter
Lens
mirror
determined the mean-free path to be ‘ ¼ 0:63 0:14 m
and the effective refractive index n ¼ 1:4 0:1 at a
free-space wavelength of 632.8 nm (See Supplemental
Material [20]).
To construct the TM, we represent the measured scattered wave fields (at different illumination angles) as columns of k-vector coefficients. The k-space representation
is generated by the two-dimensional Fourier transformation of the measured wave fields. The information is represented in a one-dimensional array whose entries are
ordered according to increasing of the lateral wave vector
modulus. This array comprises one half column in the TM
matrix. The full column consists of the measured response
for both output polarization states, corresponding to the
input mode (Fig. 2).
We make two observations about the values presented in
the matrix. Firstly, the moduli of the Fourier components of
the measured wave fields decrease with increasing modulus jkx;y j of the lateral spatial frequency. This is primarily
due to system vignetting and scattering through the side
edges of the sample which cannot be measured by the CCD
camera. These factors will have a distorting effect on the
measured singular value spectrum, in addition to that of
the limited measurement NA, as we describe below.
Secondly, the horizontal lines that are particularly visible
in the phase maps are measurement artifacts. These are due
to unwanted diffraction by dust particles in the optical
path. They can be removed in postprocessing, by subtracting the complex mean value of the matrix from each
horizontal line. However, we have found this has no discernable effect on the measured eigenvalue distribution. As
noted above, we matched the independent number of the
input modes and the output modes at 21 078. The sampled
number exceeds the resolvable number of input far-field
modes (2A=2 11 000) by a factor of about 2 for the
measured field of view of 18 18 m2 . The eigenvalues
retrieved from the measured TM were appropriately
rescaled by being divided by the oversampling ratio.
21078
CCD camera
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PHYSICAL REVIEW LETTERS
PRL 111, 153902 (2013)
input mode
10539
21078
FIG. 2 (color online). (a) Amplitude and (b) phase of the
reconstructed TM (a sampled representation of the rows and
columns is shown). Row and column numbers indicate the
spatial frequency mode indices of the outgoing and incoming
complex wave fields, respectively. Arrows represent polarization
direction.
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We investigated the statistical properties of the measured
large TM using RMT. We consider the transmission eigenvalues Ti of tty , which are equal to the squared values of
the singular values of the TM, i . According to DMPK
theory, for a highly scattering, but weakly absorbing
medium, the probability density function of Ti is given
by the equation [18,21],
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(2)
pðTi Þ ¼ hTi i=ðTi 1 Ti Þ;
" < Ti < 1:
The parameter hTi i ‘=L, is the mean transmittance of
the distribution,R and the value " ¼ cosh2 ð1=hTi iÞ is
chosen so that pðTÞdT ¼ 1 [22]. This distribution has
the important characteristic of being bimodal. A majority
of the transmission channels (corresponding to input eigenvectors) will be either ‘‘open’’ or ‘‘closed’’, corresponding
to almost-no transmission (Ti 0) and near-complete
transmission (Ti 1), respectively [23,24].
The eigenvalue distributions calculated from the measured TM are shown in Fig. 3. The average transmission
hTi i was 0.0671. A simulated TM based on RMT [25] is
also shown for the purpose of comparison. The normalized
singular value distributions are shown in Fig. 3 (inset). For
the comparison with the dimensionless theoretical predictions of RMT, the singular values i are normalized to
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PN
yield ~i ¼ i =
2j =N . The distribution from the
j¼1
simulated TM clearly exhibits the expected bimodal distribution. The measured and simulated eigenvalue distributions are reasonably matched for low values (Ti < 0:3).
However, beyond this threshold, the curves deviate
strongly from one another. In particular, the open channels
Random-Matrix Theory
2
10
1
10
0
f =1
1
Experimental data
Quarter-circle law
are not observed in the measurement. One reason for this
deviation may be the limited measurement NA, which
leads to both imperfect control of the incident field and
loss of some transmitted field information.
To test the effect of NA on the eigenvalue distribution,
we numerically incorporate the limited-NA property into
the RMT simulation. We achieve this by limiting the
number of illumination and detection modes to a fraction
f of the total number (0 < f 1) for both the illumination
and detection sides; we assume symmetry on the two sides.
Then for the ‘‘reduced’’ transmission matrix, tR entries
corresponding to either i; j > Nf are replaced with zeros.
As shown in Fig. 3, the open channels are only observed in
the ideal TM; with just 10% loss of optical modes, the open
channels completely disappear. Also noteworthy is the fact
that when f ¼ 0:65, which is the experimental condition,
the maximum values of the eigenvalue of the simulated TM
(T ¼ 0:68, i ¼ 0:82) is approximately equal to the maximum eigenvalue (0.65) obtained from the measurement.
There is a one-to-one correspondence between f and NA
through the relationship f ¼ ðNAÞ2 =n2 . To see this, consider plane wave illumination with intensity I, polar angle
with respect to the optical axis, upon a finite area A in a
plane orthogonal to the optical axis. The total power intercepted by the area is equal to IA cos. We assume that the
light scattering distribution is uniform in all directions, and
that all entries of the TM have identical (marginal) probability distributions. Then, the canonical basis vectors of
illumination and detection states (each corresponding to a
small solid angle, , of illumination or detection, respectively) must have the property that cos is constant. A
solid angle element represents surface area s of the
unit sphere. The condition ‘‘cos is constant’’ is
equivalent to the condition that the area of the projection
of s onto the equatorial plane be constant; thus, the
0.5
Random-matrix theory
both pols
both pols + defect
single pol
0.6
f =0.65
f =1
0
0
10
10
-2
2
0.5
3
λ
f =0.65 + defect
-1
1
f =0.2
f =0.65
Experiments
both pols
single pols
0.4
f =1
f =0.8
C4,2
p(T )
f =0.8
f =0.2
f =0.65
f =0.65 + defect
p(λ)
10
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PRL 111, 153902 (2013)
f max = 0.65
0.3
0.2
0
0.5
1
T
0.1
FIG. 3 (color online). Transmission eigenvalue distribution of
the measured TM. The lines represent RMT simulations with full
information (black line), information on only a fraction f ¼ 0:65
of the incident and transmitted modes (green line), and incorporation of an additional defect factor of 24% (gray line). Dashed
line represents distribution from the quarter-circle law. (Inset)
Normalized singular values distribution of the measured TM.
0
0.2
0.4
f
0.6
0.8
1
FIG. 4 (color online). Measured and ideal values for C4;2 as a
function of f. Dotted lines represent extrapolations from the
measurements.
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PHYSICAL REVIEW LETTERS
canonical basis states should be uniformly distributed over
the two-dimensional lateral spatial frequency components
of the wave field incident upon the scattering sample.
The fraction of the total circular projection area corresponding to polar angles less than a given is equal to
sin2 ¼ NA2 =n2 .
In our experiment, from the effective NA of the objectives and the index of the immersion oil, we estimate
f ¼ 0:65. However, the distribution of the simulated
eigenvalues with f ¼ 0:65 deviates from the measured one,
particularly at the high-eigenvalue end of the distribution.
This deviation suggests an additional loss of information
due to an uncollected component of the scattered field. The
possible reasons for this loss could be (1) unwanted scattering by optical components or objects such as bubbles in
the immersion oil, and (2) the aforementioned vignetting
effects and the scattering of the beam to the side of the
turbid sample, resulting in illumination that is not collected
by the objective lens. We have attempted to incorporate
this undesired information loss into our numerical simulation, by accounting for the lateral spread of the beam due to
optical diffusion. We assume that 24% of the optical output
modes were lost in the experiment. This value was calculated as follows. All incident modes cover the sample area
of 18 18 m2 . However, the average illumination profile on the output side will be expanded by a factor of
approximately 1.31 (in area), the quantity being determined pvia
ffiffiffi convolution of the original spot profile with
sechðx 6=lÞ [26]. Thus, 24% (0:31=1:31) of the light
and corresponding optical modes are lost in the experiments. To simulate this ‘‘defect’’, 24% of the output modes
(randomly chosen) are replaced with zeros (gray line in
Fig. 3), in a similar manner to the construction of the
reduced transmission matrices, described above. The
resulting RMT simulation shows improved agreement
with the measured eigenvalue distribution.
The singular values i of a large random matrix with
independent, identically distributed entries will follow a
quarter-circle distribution [27]. If correlations between the
entries exist in our measured TM, we expect a deviation
from this quarter-circle limit. The distribution for Ti corresponding to a quarter-circle law for i is plotted in Fig. 3,
clearly showing a different form to that of our experiment.
By contrast, Popoff and co-workers [15] measured a relatively small fraction of the TM (we estimate f < 0:001 for
their situation) and found a singular value distribution that
nicely followed the quarter-circle law, indicating all correlations were lost from the matrix.
An important characteristic of the distribution pðTi Þ is
C4;2 , the ratio of itsRsecond and first
R moments. Theoretically,
C4;2 ¼ C4 =C2 ¼ T 2 pðTi ÞdT= TpðTi ÞdT ¼ 2=3. This
value of C4;2 is universal, i.e., independent of hTi i or
sample thickness. Recently, it has been demonstrated that
TM can be used for focusing light [28] and image delivering [1,2]. From a theoretical point of view, the fraction C4;2
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gives the proportion of the incident power that is transmitted (on average) when wave front control of the incident beam is used to optimize the intensity at a point
behind the turbid medium, regardless of the thickness of
the sample [28]. One advantage of C4;2 is that it can be
directly estimated from the Hermitian matrix H ¼ ðhi;j Þ tty , without requiring that the eigenvalues be individually
calculated. In the Supplemental Material [20], we show
that the expected value of C4;2 corresponding to a reduced
TM is directly proportional to NA2D NA2I =n4 , where NA2D
and NA2I represent NA for the detection and illumination
side, respectively. See Supplemental Material [20] for the
derivation.
To investigate experimentally the effect of limited NA
on C4;2 , we calculated its estimate C^ 4;2 from the measured
TM (Fig. 4). For the maximum value of f, the calculated
C^ 4;2 values are 0.180 (full polarization-dependent TM) and
0.117 (only one linearly polarization dependent for both
input and output modes). These values are consistent with
the report [28], where the maximum transmittance of an
optimized beam was 0:2 with a high-NA objective lens, a
scattering sample with comparable optical properties, and
the light polarization was considered.
To further address the effect of NA upon the values of
C4;2 , we extrapolated the values of C^ 4;2 for larger values of
f than those permitted experimentally (dotted lines in
Fig. 4). The extrapolation was obtained using the
MatlabÒ fitting toolbox. The extrapolations for the cases
of a single polarization and two polarizations showed
trends similar to RMT predictions. The maximum values
of the extrapolated C^ 4;2 (for f ¼ 1) was 0.32 (4% smaller
than that expected from RMT) for the case of the unpolarized beam. When all polarization components were taken
into account, the extrapolated C^ 4;2 was 0.52, which is 22%
smaller than expected from RMT. We attribute this deviation in the values of C4;2 between the measurement and
RMT mainly to the leakage of light to the side of the
sample. When we simulate the values of C4;2 considering
24% leakage in the TM, we obtained results that are
comparable with those of the measurement.
In conclusion, we have demonstrated that large optical
TM of a turbid medium can be experimentally measured
using a fast two-axis galvanometer mirror and full-field
interferometric microscopy equipped with a high-NA
objective lens. Our work provides a means for directly
measuring the TM and leads to observation of the intrinsic
correlation between the modes of the measured optical
TM. The total number of entries in the measured TM is
approximately 420 million, which is slightly more than
the number of resolvable modes for measured area of the
18 18 m2 field of view. Both the transmission eigenvalue and singular value distribution obtained from the
measured TM deviate from the quarter-circle shape characteristic of a matrix with independent, random entries.
This is in satisfactory agreement with the distribution
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PHYSICAL REVIEW LETTERS
predicted theoretically, and simulated numerically, when
corrected for the limited illumination and detection NAs.
Our measurement method can be applied in situations
where large transmission matrices are needed and will
open new possibilities in focusing and imaging applications and will have many potential applications
[1,2,7,29,30]. Furthermore, the present approach can be
further expanded to the measurement of reflection matrices
as well as scattering matrices. Moreover, localized optical
modes [10,31] or modes of the evanescent near field resulting from multiple scattering of nanoparticles [32] may be
accessed using measurement of the large TM.
The authors thank Elbert van Putten and Allard Mosk
methods, for their sample preparation during the early
stages of the project, and for proofreading the manuscript. The authors also wish to acknowledge Dr. Zahid
Yaqoob for helpful discussions. This research was
supported by the National Institutes of Health (9P41EB015871-26A1), the National Science Foundation
(DBI-0754339), KAIST, MEST/NRF (2009-0087691
(BRL), 2012R1A1A1009082, 2012K1A3A1A09055128,
2012-M3C1A1-048860, 2013M3C1A3063046), and
Hamamatsu Photonics, Japan. Y. K. P. acknowledges support from TJ ChungAm Foundation. H. Y. and T. R. H.
contributed equally to this work.
Note added.—After submission of this Letter, a relevant
theoretical paper appeared [33].
*Deceased.
†
yk.park@kaist.ac.kr
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